On the Statistical Foundations of H-likelihood for Unobserved Random Variables

The maximum likelihood estimation is widely used for statistical inferences. In this study, we reformulate the h-likelihood proposed by Lee and Nelder in 1996, whose maximization yields maximum likelihood estimators for fixed parameters and asymptotically best unbiased predictors for random parameters. We establish the statistical foundations for h-likelihood theories, which extend classical likelihood theories to embrace broad classes of statistical models with random parameters. The maximum h-likelihood estimators asymptotically achieve the generalized Cramer-Rao lower bound. Furthermore, we explore asymptotic theory when the consistency of either fixed parameter estimation or random parameter prediction is violated. The introduction of this new h-likelihood framework enables likelihood theories to cover inferences for a much broader class of models, while also providing computationally efficient fitting algorithms to give asymptotically optimal estimators for fixed parameters and predictors for random parameters.